* Did you have supper? - Asking if the person has attended supper. * Did you have supper yet? - As far as I know , that`s grammatically not correct. * Have you had supper yet? - Asking if the person will attend supper.
Did you have supper yet?-- 'Did you eat before now?-- because I may invite you to supper.' (Yet & already are common with simple past in AmE at least. Yet is used in negative statements and questions.)